#503046

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

749 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(5,6,8,9)(7,11,12,10)(13,14)(15,25)(16,30)(17,22)(18,37)(19,38)(20,27)(21,33)(23,31)(24,36)(26,40)(28,44)(29,39)(32,41)(34,43)(35,42);(1,2)(3,4)(6,9)(10,11)(13,15)(14,20)(16,23)(17,27)(18,28)(19,29)(21,36)(22,25)(24,39)(26,32)(30,40)(31,41)(33,38)(34,37)(35,43)(42,44);(2,4)(13,16,31,20,19,34,35,39,17,32,40,25,36,44,18,33)(14,21,37,28,24,15,26,41,22,29,42,43,38,27,23,30);(5,7,8,12)(6,10,9,11)(14,22)(15,27)(21,29)(23,26)(24,38)(28,43)(30,41)(37,42)

Polycyclic presentation:

2^9|a^2=ei,c^2=fh,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^b=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410242);